Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C22⋊C4 |
| commutator | C22 | C2×C4×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | C16×C22⋊C4 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,5,11,6,13,25,41,7,14,26,42,28,45,65,86,8,12,27,40,29,43,66,84,30,44,67,85,68,87,105,117)(3,15,18,47,19,48,56,91,20,49,57,92,58,93,100,121,17,9,31,33,54,34,70,75,55,35,71,76,99,77,108,112)(4,10,21,36,22,38,59,79,23,39,60,80,62,83,101,115,24,37,61,78,63,81,102,113,64,82,103,114,104,116,124,127)(16,46,51,88,52,89,96,118,53,90,97,119,98,120,123,128,50,32,69,72,94,73,106,109,95,74,107,110,122,111,125,126);(1,3)(2,9,12,15)(4,16)(5,18)(6,19)(7,20)(8,17)(10,32,37,46)(11,33,40,47)(13,34,43,48)(14,35,44,49)(21,51)(22,52)(23,53)(24,50)(25,56)(26,57)(27,31)(28,58)(29,54)(30,55)(36,72,78,88)(38,73,81,89)(39,74,82,90)(41,75,84,91)(42,76,85,92)(45,77,87,93)(59,96)(60,97)(61,69)(62,98)(63,94)(64,95)(65,100)(66,70)(67,71)(68,99)(79,109,113,118)(80,110,114,119)(83,111,116,120)(86,112,117,121)(101,123)(102,106)(103,107)(104,122)(105,108)(115,126,127,128)(124,125);(1,4,5,21,6,22,25,59,7,23,26,60,28,62,65,101,8,24,27,61,29,63,66,102,30,64,67,103,68,104,105,124)(2,10,11,36,13,38,41,79,14,39,42,80,45,83,86,115,12,37,40,78,43,81,84,113,44,82,85,114,87,116,117,127)(3,16,18,51,19,52,56,96,20,53,57,97,58,98,100,123,17,50,31,69,54,94,70,106,55,95,71,107,99,122,108,125)(9,32,33,72,34,73,75,109,35,74,76,110,77,111,112,126,15,46,47,88,48,89,91,118,49,90,92,119,93,120,121,128)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,e^2=g,g^2=h,h^2=i,b^a=bd,f^a=fi,d^b=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59095);